#27083549

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3057 ⋊ C₂
Sylow	C₄₇ ⋊ #60455

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3057	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20256719	C₂
hypercenter	#3057	D₄₇
hypocenter	C₄₇	#60455
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #3057

2678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 83 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(54,64)(58,72)(62,76)(63,77)(65,70)(71,82)(75,83)(78,81);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,66)(59,73)(63,77)(67,79)(71,82)(74,80)(75,83)(78,81);(49,51)(52,55)(53,59)(54,63)(56,67)(57,66)(58,71)(60,74)(61,73)(62,75)(64,77)(65,78)(68,80)(69,79)(70,81)(72,82)(76,83);(52,54,56,65,57,64,69,70)(53,58,60,62,61,72,68,76)(55,63,67,78,66,77,79,81)(59,71,74,75,73,82,80,83);(48,49)(50,51)(52,53,57,61)(54,62,64,76)(55,59,66,73)(56,68,69,60)(58,70,72,65)(63,75,77,83)(67,80,79,74)(71,81,82,78)

Polycyclic presentation:

2^8,47|a^2=h,b^2=f,f^2=h,b^a=bf,c^a=cg,f^a=fh,e^b=eh,d^c=dh,i^e=i^46

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036839);